1. Field of the Invention
The present invention relates to a method of optimizing an optical system which realizes automated design of an optical system, an apparatus for optimizing an optical system and a recording medium on which a program for optimizing an optical system is recorded.
2. Description of the Related Art
In recent years, it has been common to employ automated design using a computer for design of optical systems (simply referred to as “optical design” hereinafter). The automated optical design is realized by executing a design program on a computer. Thus, in performing the automated optical design, use of excellent design programs is an important factor to realize favorable optical design.
Optical design is an operation to optimize design parameters of an optical system so that the optical system offers desired optical performance. The basic concept of the optimization of an optical system will be described herein below. Although the “optical system” denotes a system including at least one optical element such as a lens or a mirror, described hereinafter is a case where the optical system comprises only lenses, for a simple description. However, an optical design including other optical elements than the lens is realized basically in the same manner as the lens design described below.
In the field of lens design, various numeric values indicating the configuration of a lens such as curvature of each lens surface, surface separation, refractive index of glass, dispersion, aspheric surface coefficient are called parameters. At least one of optical characteristics of a lens such as paraxial amounts (focal length, back focus and the like), aberration and lens shape is considered to evaluate the lens performance and a target value is set for each of evaluation subjects (i.e., optical characteristics to be evaluated). These parameters are changed in such a manner that the evaluated value of each optical characteristic becomes as close to the given target value as possible.
In the field of lens design, a function for evaluation of lenses is generally called “merit function” (or evaluation function). The merit function is a sum total of functions represented by wi*(fi−ti)2 of all evaluation subjects. In the wi*(fi−ti)2, ‘fi’ means a temporary characteristic value (evaluated value) of the ith evaluation subject (i represents an integer greater than or equal to 1); ‘ti’ means a target value of the ith evaluation subject; ‘wi’ means a weight value of the ith subject; and ‘*’ means multiplication. To obtain the parameter values that make the merit function the minimum is the essential operation of the automated lens design. There are many algorithms for minimizing the merit function such as a damped least squares (DLS) method and a quasi-Newton method. It is ideal that all evaluation values fi is equal to the target values ti, but practically it is not always. When all evaluation values do not correspond to the target value ti, varying balance between each weight wi corresponding to each target value ti changes in turn the minimum of the merit function. Operation to seek the minimum point of the merit function by varying parameters is generally called “lens optimization”.
When designing lenses, design conditions such as paraxial amounts, e.g., focal length and back focus, lens shape, e.g., length or diameter of lens, tolerance level of distortion are required to be set identical to specifications. The specifications are given as a specific value or a value with a tolerance to a degree. Here, the design condition in which values are identical to the specification is called “constraint”. The optimal point within a parameter space in lens optimization is formulated to be the minimum of the merit function under the conditions where all constraints are identical to the specifications.
It is necessary for lenses to have preferable performance in image formation. For instance, an image pickup lens has to have preferable image formation performance within the range to be used of an object distance and the size of the screen. Especially, when the image pickup lens is a zoom lens, preferable image formation performance should be ensured within the range of an object distance and the size of the screen over the whole zooming range. In the design of such a lens, first of all, several typical zoom points are selected in the zooming range and optical performance at each of the zoom points is subject to optimization. Then, several typical object distances at each zoom point are selected and several typical points are further selected on the object plane. Thus, the zoom lens is comprehensively optimized as to the state of image formation on the image plane, the image formation being performed by rays from a plurality of sample object points, the rays being sampled over several zoom points and object distances. Although the image pickup lens having a zoom function has been described above as an example, the above-mentioned process is not limited to this particular lens. General operation of the lens design is an optimization of image formation on the image plane of one or more of sample object points selected in accordance with a particular application of a lens.
Typical optical properties used in the evaluation of lens performance will be now described. In an optical system, a light from an object point passes through a lens and the like and reaches on the image plane in the form of a spread flux. The flux reached on the image plane forms intensity distribution thereon in accordance with the properties of the optical system, which is called a point spread function (PSF). In a preferable state of image formation, the PSF takes a value other than zero in a very small range. Fourier transform of the PSF on condition that integration of intensity over the whole of image plane is normalized as ‘1’ is called an optical transfer function (OTF). The OTF is a complex function in a two dimensional frequency space. The absolute value of the OTF is called a modulation transfer function (MTF). The MTF takes on values between 0 and 1 and always 1 at the origin of the frequency space. The bigger the value of the MTF is, the better the image formation of the optical system is. The target in a lens performance is often set using the MTF. The MTF is generally evaluated in respect to a sagittal direction (S direction) and a tangential (or meridional) direction (T direction). Here, S direction denotes a direction vertical to a meridional plane, while T direction denotes a direction included in the meridional plane. The “meridional plane” denotes a plane including both an axis of symmetry and an object point in an axisymmetric lens when the object point is not on the axis of symmetry.
Several sample rays are selected from a flux reaching from the object point to the image plane and positions of these rays on the image plane are plotted, and a thus formed diagram is called a spot diagram. The more these rays are converged on one point, the better image formation of the optical system is. Two axes orthogonal to each other on the image plane are designated as an X coordinate axis and a Y coordinate axis. Xi and Yi are coordinates of the ith sample ray on the image plane, and Xm and Ym are the mean of coordinates Xi and Yi over all sample rays, respectively. Root of mean of (Xi−Xm)2+(Yi−Ym)2 over ‘i’ is called a root mean square (RMS) spot size. The RMS spot size is a numeric value showing a state how rays are converged. When the value is small, the spread of the PSF is small. When the light source is not a monochromatic light, several sample wavelengths are chosen, and ray tracing for each sample wavelength is conducted, in general.
A ray with a standard wavelength passing through the center of the stop of an optical system is called a “principal ray”. When Xc and Yc are coordinates on the image plane of the principal ray and Xi and Yi are coordinates on the image plane of the ith sample ray, Xi−Xc and Yi−Yc are called “transverse aberration” of the sample ray when taking the principal ray as a reference. This transverse aberration is a typical subject of the performance evaluation in the automated lens design. In one typical example of the merit function, the transverse aberration with regard to all sample rays from all sample object points is subjected to evaluate the performance. In the case where transverse aberration is the evaluation subject in the merit function, the simplest setting for the target value and weight is to set the target value to ‘0’ and the weight to ‘1’ with respect to X components and Y components of all sample rays from all sample object points. Considering that there is not much differences between the coordinates Xc and Yc of the principle ray and the mean coordinates Xm and Ym of all sample rays, the merit function on transverse aberration corresponds to a merit function acquired by summing root squares of the RMS spot size over all sample objects. In the automated lens design, the target value of transverse aberration of all sample rays is, however, not necessarily to be the same. The same weight also does not need to be set with respect to all sample rays. The target value and weight can be arbitrarily set with respect to each sample ray. When the target value and weight of each sample rays are changed, an optimal solution of the automated design changes.
In addition to the aberration described above, other examples of lens aberration are transverse aberration taking a paraxial image point as a reference, wavefront aberration, spherical aberration, curvature of field by a principal ray and the like. Use of these aberrations as the evaluation subject in the merit function is effective to optimize the image formation.
Meanwhile, in the automated lens design using the merit function in view of only general aberration, although the optimal solution is effective for the evaluation of aberration, other optical performance such as an evaluation of the MTF is not always optimized. When values of the MTF is given as a target of lens performance, lens optimization may be performed using the merit function on the MTF. However, there exist the following problems in that case.
The MTF has higher non-linearity with parameters than that of aberration, thus it is difficult to optimize lenses effectively as compared to the case where aberration is subjected to be evaluated. It is apparent that the MTF has high non-linearity from the fact that the MTF takes on values only between 0 and 1. Meanwhile, since a greater value of the MTF is preferable, it is natural to set the design target to maximize a value obtained by summing all MTF values as the performance objective (e.g., a value obtained by summing all MTF values at each field angle). In this case, the merit function subjected to be maximized is a linear combination on the MTF. However, conventional algorithms for optimization such as a DLS method are assumed to be used on an evaluation subject with low non-linearity such as aberration, and objective thereof is to minimize the sum total of squares of the differences between the evaluated values and the target values. Thus, the conventional algorithms for lens optimization are inappropriate for optimization on the evaluation subject having high non-linearity such as the MTF. As described above, with conventional optimization techniques, it is difficult to efficiently realize lens optimization on the MTF.
There is another problem such that calculation for the MTF takes longer ti me than that for aberration. When optimization with aberration as the evaluation target is performed, the number of sample rays at one sample object point is about 10 to 20. To calculate the MTF with practical accuracy, however, at least more than 100 of rays have to be traced at one sample object point. Further, a process for obtaining the MTF from the result of ray tracing is complicated compared to that for obtaining aberration, resulting in requiring further longer time. As described above, it is apparent that lens optimization on the MTF takes long time.
As described above, lens optimization method using the merit function on the MTF has various problems. Therefore, practically repetition of the following operation is common in lens design. First, an optimal solution of the lens on aberration is obtained. Then, the MTF is evaluated with regard to the optimal solution. A designer adjusts weights and target values on aberration in order to obtain better values of the MTF and re-optimizes the lens design on aberration. This series of steps is repeated. In such a lens design, however, manual work by a designer is required, so that high-speed calculation by a computer, which is an advantage of the automated design, cannot be fully utilized. Adjusting weights or target values for aberration to obtain a preferable MTF values needs experiences of a designer. Therefore, it is considered that if the function equivalent to the operation of manually adjusting weights or target values by a designer is automated, efficient and high-speed lens design can be realized for the purpose of optimizing the lens on the MTF.
This invention is achieved with a view of above problems. It is an object of the invention to provide a method of optimizing an optical system in which optical properties with high non-linearity such as the MTF are optimized at high speed compared to conventional methods, an apparatus for optimizing an optical system and a recording medium on which an optimization program of an optical system is recorded.